flightaware.com track log shows the flight starting off with atlanta center and then most of it with jacksonville center. I know that's not the most accurate function on that website, but it sounds reasonable.
I can say 100 percent for sure that you are correct that the tracklogs on Flight Aware are not accurate with regards to what ATC component you talk with.
My flights to KEKY, I talk with ATL center, switch over to KBMH and the track log will not reflect anything. Track line drops where my handoff to BMH approach begins. (see below)
http://flightaware.com/live/flight/N1943L/history/20081127/1240Z/KMBO/KEKYhttp://flightaware.com/live/flight/N1943L/history/20081127/1243Z/KMBO/KEKY/tracklogOn my return, I get transferred from Memphis Center and talk with KJAN approach and my tracklogs will only show the center frequency even though the trackline does show my track while I am under KJAN. On the below flight, for the GPS approach I was with KJAN.
http://flightaware.com/live/flight/N1943L/history/20081127/2140Z/KEKY/KMBOhttp://flightaware.com/live/flight/N1943L/history/20081127/2131Z/KEKY/KMBO/tracklogI was told by Flight Aware Staff members that BMH most likely marks me arrived, thus my tracking stops. This does not happen at KJAN.